Module 1 : General Topology Lecture 6 :
(Test - I)
1. Prove that the intervals and are non-homeomorphic subsets of . Prove that if and are homeomorphic subsets of , then is open in if and only if
is open in . Is an injective continuous map a homeomorphism onto its image?
2. Using Tietze's extension theorem or otherwise construct a continuous map from into such that the image of is not closed in .
3. If is a compact subset of a topological group and is a closed subset of , is it true that is closed in ? What if and are merely closed subsets of ? 4. Removing three points from we get an open set and a continuous map
given by . Which three
points need to be removed? Prove the continuity of .
5. Let . Is connected? Is
homeomorphic to ?
6. Prove that is homeomorphic to .
